Inverse-free Solovay–Kitaev theorem

Let $d\geq 2$ be fixed, and let $\mathcal{G}\subset \mathrm{SU}(d)$ be a finite gate set that densely generates $\mathrm{SU}(d)$. An inverse-free Solovay–Kitaev theorem asserts that there is an algorithm which, for every $U\in \mathrm{SU}(d)$ and every $\epsilon>0$, outputs an $\epsilon$-approximation to $U$ using merely

$$O(\operatorname{polylog}(1/\epsilon))$$

elements from $\mathcal{G}$. The generic inverse-free version would extend Solovay–Kitaev compilation to arbitrary finite dense gate sets without requiring inverses; proving it is described as the main unresolved problem left by the work and a longstanding open problem.